A concept I'd like to do is have a character gain a specific status effect, Wounded, on reaching 0 hp. Then, if they reach 0 hp again while Wounded, they die. A character that's wounded can be healed but suffers penalties until they are. A dead character is removed from the game and gone forever.
Does the engine allow for this?

What I'd do here is to use HP to merely present the state wounded or not. So every hero just has 2 Max HP. When he's at 1 HP it's like he's wounded and when he's at 0 HP he's dead. An after battle plotscript can remove any hero with 0 HP from the party.

Then you'd need to design skills that can only deal 1 HP damage at max but they only deal damage on certain other conditions.

There are various ways you can go about the conditions. One would be to use the DEF stat so that DEF actually works like HP. Attacks reduce DEF and then deal HP damage that is reduced by DEF. Their base damage is 1 and is reduced by DEF so as long as DEF is higher than 1, the HP damage will be 0.

An alternative would be to deal damage through skill chaining and then you can put all the logic in the skill chain condition. "Only chain to 1 HP damage skill if target stat is 0".

Yes, you can do that, but instead of changing the meaning of the HP and DEF skills there's a simpler way using chaining: use the CTR stat (which does nothing, so you're free to use it for whatever you want) to indicate 'Wounded". Lets say 0 means OK and 1 means wounded

Then as Rya said, make every attack chain to a special sequence of attacks:
-if HP has reached 0 then chain:
-if CTR is 0 (not wounded) then chain:
-heal to full HP and chain:
-increase CTR by 1
-chain to further attacks to cause any debuffs you want

Unfortunately this doesn't work for multi-target attacks_________________"It is so great it is insanely great."